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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Rank of a Matrix

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The linear system M‾⋅x⃗=0⃗\bcb{\boldsymbol{ \underline{M} \cdot \vx = \vz}} is known to have only the trivial solution. If M‾\bcb{\boldsymbol{ \underline{M}}} is a matrix with 8 rows and 7 columns, what is its rank?
More Rank of a Matrix Questions:
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
Example: Rank of a Matrix
Example:
1. Find the ranks of the following matrices and state the number of solutions.
a.) [102100∣120]\left[\begin{array}{l} 1& 0 \\ 2& 1 \\ 0 & 0 \\ \end{array}\right.\left|\begin{array}{l} 1\\2\\0\end{array}\right]
The row echelon form of the augmented matrix for a linear system is:
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
What is the possible rank of an augmented matrix of five linear equations with four unknowns if the solution is a plane.
A linear system of three equations in four unknowns has
The reduced row echelon form of the augmented matrix for a linear system is
[A∣b][10111012400000100000∣1258].[A|b]\left[\begin{array}{l} 1&0&1&1&1\\ 0&1&2&4&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{array}\right.\left|\begin{array}{l}1 \\2\\5\\8\end{array}\right]_.
a) What is the rank of matrix A
Equivalent Conditions to Full Rank
a) If AA is a 5×35\times 3 matrix of full rank, and BB is a 3×73\times 7 matrix of rank 2, what are the possible values of rank(AB)rank(AB) ?
b) If AA is a 3×53\times 5 matrix of full rank and BB is a 5×45\times 4 matrix of rank 3, what are the possible values of rank(AB)rank(AB) ?
Find the rank of the following matrix:
[525151021200]\begin{bmatrix}5&2&5&1\\5&1&0&2\\1&2&0&0\end{bmatrix}
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b]\left[A\left|b\right.\right]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AAis 3, which of the following statements best describes the system?
Suppose that a system of mmlinear equations in nnvariables is represented by the augmented matrix [A∣b][A|b]. If the system is consistent, which of the following statements is always true?
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AA is rr, what is the maximum value rrcan be?
Given a system of 3 linear equations in 5 variables, if the rank of the coefficient matrix AA is 3, which of the following statements best describes the system?
Linear equations
Given a system of 8 linear equations in 5 variables, if the rank of the coefficient matrix AAis rr, what is the maximum value rrcan be?
Matrix of a Projection
a) Find the matrix AA so that for x⃗∈R3\vec x \in \mathbb{R}^3 the transformation mapping x⃗→proj(−13,1,12)x⃗\vec x\rightarrow proj_{(\frac{-1}{3},1,\frac{1}{2})}\vec x is the same as the transformation mapping x⃗→Ax⃗\vec x\rightarrow A\vec x
b) What is rank(A)rank(A) ?
Invertible Matrix Theorem
Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏} where A5×5A_{5 \times 5} is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])?
Rank of a Matrix
Determine the number of solutions for the system of linear equations Ax⃗=b⃗A\vec{x}=\vec{b} with 5 equations and 5 unknowns, given:
Rank of a Matrix
In the reduced row echelon form of the coefficient matrix AA that corresponds to a system of linear equations with 6 equations in 9 unknowns, the maximum number of leading 1s is
6
.
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a matrix
Let AA be a 6 x 4 matrix and let Ax=bA\bm{x}=\bm{b} be a system of linear equations. Which of the following statements must always be true? (select all that apply)
Rank of a Matrix
Determine the rank of [1−10320131110]\begin{bmatrix} 1&-1&0&3\\ 2&0&1&3\\ 1&1&1&0 \end{bmatrix}
Rank of a Matrix
How many solutions does the system of linear equations corresponding to the following augmented matrix have?
[1010−1101101000011−1000000]\left[ \begin{array}{rrrrr|r} 1&0&1&0&-1&1\\ 0&1&1&0&1&0\\ 0&0&0&1&1&-1\\ 0&0&0&0&0&0 \end{array} \right]
Rank of a Matrix
Which of the following is an augmented matrix corresponding to a system of linear equation that is consistent? (select all that apply)